Finite Group Databasebeta

← C2×#9321 #199333

Subgroups — C2×#9321 #199333

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2C2×#2989C211yesmaximal, characteristic
3C2×#1981C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic, complemented
6#9321C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic, complemented
8C2×#1292C211yesmaximal, characteristic, complemented
9C22×C4⋊Q8C2211yescharacteristic
10C2×C4×#41C2211yescharacteristic
11#3079—22no—
12#3079—22no—
13#3079C414yescomplemented
14#3079C2214yes—
15C2×#477C412yescomplemented
16C2×#475—21no—
17C2×#476C412yescomplemented
18C2×#474—21no—
19#2989—22no—
20#2989—22no—
21#2989—22no—
22#2989C2214yes—
23C2×#446—21no—
24C2×#436—21no—
25C2×#435—22no—
26C2×#432—21no—
27C2×#431C2211yescharacteristic
28C2×#428—21no—
29C2×#423—21no—
30C2×#423C2211yescharacteristic
31C2×#423C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#104—21no—
34C2×#417—21no—
35C22×#103—22no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1981—22no—
38#1981C2214yescomplemented
39#1806—22no—
40#1806C2214yes—
41C2×#339—21no—
42#1549—22no—
43#1549C2214yescomplemented
44C2×#332—21no—
45C2×#332—21no—
46#1303C2214yescomplemented
47#1292C2214yescomplemented
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C22×M16—21no—
51C22×M16—22no—
52C22×M16—21no—
53C23×C8D412yes—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8C2×C412yes—
65C2×C4⋊Q8C2312yes—
66C4×#41—21no—
67C4×#41—21no—
68C4×#41—21no—
69C4×#41—21no—
70C4×#41C2×C412yes—
71C4×#41D412yes—
72C4×#41D412yes—
73C4×#41C2312yes—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×#117—21no—
81C2×#117D411yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#477—24no—
89#477C2×C418yescomplemented
90#476—24no—
91#476C2×C418yescomplemented
92#475—24no—
93#475—24no—
94#474—24no—
95#474—24no—
96#446—24no—
97#446—24no—
98#436—24no—
99#436—24no—
100#435—28no—
101#435—28no—
102#432—24no—
103#432—24no—
104#431—21no—
105#431—21no—
106#431D412yes—
107#431C2312yes—
108#428—24no—
109#428—24no—
110C2×#107—22no—
111C2×#106—22no—
112#423—24no—
113#423—21no—
114#423—21no—
115#423—24no—
116#423—21no—
117#423—21no—
118#423—21no—
119#423C2×C412yes—
120#423C2312yes—
121#423C2312yes—
122C2×C4⋊C8—22no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8C2×C412yes—
131C2×C4⋊C8D412yes—
132C2×C4⋊C8D411yescharacteristic
133C2×C4⋊C8D411yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×#104—24no—
136C2×#104—44no—
137#417—24no—
138#417—24no—
139C2×#103—22no—
140C2×#103—28no—
141C2×#103—48no—
142C2×#102—22no—
143C2×C22⋊C8—22no—
144C2×C22⋊C8—21no—
145C2×#97—22no—
146C2×C4×C8—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4C2312yes—
151C2×#95—21no—
152#339—24no—
153#332—24no—
154#332—24no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41C22⋊C412yes—
162C2×#41C2×D412yes—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8C22⋊C411yescharacteristic
172C22×D4—21no—
173C22×D4C22⋊C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—41no—
178C23×C4—41no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—28no—
185C2×M16—22no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16—41no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—84no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—44no—
198C22×C8—41no—
199C22×C8C2×D414yes—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209C4⋊Q8—41no—
210C4⋊Q8—41no—
211C4⋊Q8—41no—
212C4⋊Q8—41no—
213C4⋊Q8—41no—
214C4⋊Q8—41no—
215C4⋊Q8C22×C414yes—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8C22⋊C412yes—
221C4×Q8C22⋊C412yes—
222C4×Q8C22×C412yes—
223C4×Q8C2×D412yes—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4C22⋊C412yes—
237C4×D4C22⋊C412yes—
238C4×D4C22×C412yes—
239C4×D4C2×D412yes—
240#117—21no—
241#117—21no—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117C22⋊C412yes—
247#117C2×D412yes—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—41no—
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42C2411yescharacteristic
294#107—28no—
295#106—28no—
296C4⋊C8—28no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300C4⋊C8—22no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—41no—
309C4⋊C8—41no—
310C4⋊C8C22×C414yes—
311C4⋊C8C2×D412yes—
312C4⋊C8C2×D412yes—
313#104—48no—
314#103—28no—
315#103—416no—
316#102—28no—
317C22⋊C8—28no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320#97—28no—
321C4×C8—22no—
322C4×C8—22no—
323#95—22no—
324#95—41no—
325C24—21no—
326C24#12111yescharacteristic
327#41—22no—
328#41—22no—
329#41—22no—
330#41—22no—
331#41—24no—
332#41—24no—
333#41—42no—
334#41—42no—
335#41—42no—
336#41—42no—
337C2×Q8—21no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—44no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—44no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—84no—
349C2×Q8—82no—
350C2×Q8—82no—
351C2×Q8—82no—
352C2×Q8—82no—
353C2×Q8C2×C22⋊C412yes—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4C2×C22⋊C412yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—84no—
386C22×C4—82no—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4#12011yescharacteristic
394C22×C4#12011yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12111yescharacteristic
397C22×C4#12111yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12311yescharacteristic
401C22×C4#12311yescharacteristic
402C22×C4AGL(1,ℤ/8)12yes—
403C22×C4#13712yes—
404C22×C4#13712yes—
405C22×C4#13712yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409M16—42no—
410M16—42no—
411M16—44no—
412M16—42no—
413M16—42no—
414M16—88no—
415M16—82no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—44no—
425C2×C8—42no—
426C2×C8—82no—
427C2×C8—82no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4C2×C22⋊C412yes—
446C4⋊C4C2×C22⋊C412yes—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—42no—
460C42—42no—
461C42—82no—
462C42C2×C22⋊C411yescharacteristic
463C42C2×C22⋊C411yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42C4×D411yescharacteristic
467C42C4×D411yescharacteristic
468C42#12011yescharacteristic
469C42#12011yescharacteristic
470C42#12111yescharacteristic
471C42#12111yescharacteristic
472C42#12311yescharacteristic
473C42#12311yescharacteristic
474C42C23×C411yescharacteristic
475C42C22×D411yescharacteristic
476C42C22×D411yescharacteristic
477C42C2×#4111yescharacteristic
478C23—22no—
479C23—22no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23#38611yescharacteristic
485C23#38611yescharacteristic
486C23#46812yes—
487C23#47012yes—
488C23C2×#12111yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—48no—
492Q8—42no—
493Q8—84no—
494Q8—84no—
495Q8—82no—
496Q8—82no—
497D4—22no—
498D4—22no—
499D4—44no—
500D4—42no—
501D4—42no—
502D4—42no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—88no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—82no—
538C2×C4#38611yescharacteristic
539C2×C4#38611yescharacteristic
540C2×C4#38611yescharacteristic
541C2×C4#38611yescharacteristic
542C2×C4#38611yescharacteristic
543C2×C4#38611yescharacteristic
544C2×C4#41712yes—
545C2×C4#41712yes—
546C2×C4#41912yes—
547C2×C4#41912yes—
548C2×C4#45312yes—
549C2×C4#45412yes—
550C2×C4#46812yes—
551C2×C4#46812yes—
552C2×C4#46812yes—
553C2×C4#47012yes—
554C2×C4#47012yes—
555C2×C4#47012yes—
556C2×C4C22×C22⋊C411yescharacteristic
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C2×#12011yescharacteristic
560C2×C4C2×#12111yescharacteristic
561C2×C4C2×#12311yescharacteristic
562C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
563C2×C4C2×#13711yescharacteristic
564C2×C4C2×#13711yescharacteristic
565C2×C4C2×#13711yescharacteristic
566C8—44no—
567C8—44no—
568C8—42no—
569C8—42no—
570C8—82no—
571C22—21no—
572C22—21no—
573C22—24no—
574C22—24no—
575C22—21no—
576C22—21no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—84no—
581C22#188312yes—
582C22#189412yes—
583C22#189412yes—
584C22#190212yes—
585C22C2×#38611yescharacteristic
586C22C2×#46811yescharacteristic
587C22C2×#47011yescharacteristic
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—84no—
598C4—84no—
599C4#188312yes—
600C4#189412yes—
601C4#189412yes—
602C4C2×#41911yescharacteristic
603C4C2×#41711yescharacteristic
604C4C2×#45311yescharacteristic
605C4C2×#45411yescharacteristic
606C4C2×#46811yescharacteristic
607C4C2×#47011yescharacteristic
608C2—21no—
609C2—22no—
610C2—21no—
611C2—82no—
612C2#932114yesdirect product factor
613C2C2×#188311yescharacteristic
614C2C2×#189411yescharacteristic
615C2C2×#190211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#1981C2semidirect
5. C2×#1549C2semidirect
6. #9321C2direct (N × H)
7. C2×#1303C2semidirect
8. C2×#1292C2semidirect
13. #3079C4semidirect
15. C2×#477C4semidirect
17. C2×#476C4semidirect
38. #1981C22semidirect
43. #1549C22semidirect
46. #1303C22semidirect
47. #1292C22semidirect
89. #477C2×C4semidirect
91. #476C2×C4semidirect
612. C2#9321direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#30792
C2×#29892
C2×#19812
C2×#18062
C2×#15492
#93212
C2×#13032
C2×#12922